Derivative of ln((2x+1)/(x-3))

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   /2*x + 1\
log|-------|
   \ x - 3 /

$$\log{\left(\frac{2 x + 1}{x - 3} \right)}$$

log((2*x + 1)/(x - 3))

Detail solution

Let $u = \frac{2 x + 1}{x - 3}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{2 x + 1}{x - 3}$ :
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = 2 x + 1$ and $g{\left(x \right)} = x - 3$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Differentiate $2 x + 1$ term by term:
    1. The derivative of the constant $1$ is zero.
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $2$
    The result is: $2$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $x - 3$ term by term:
    1. The derivative of the constant $-3$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  Now plug in to the quotient rule:
  
  $- \frac{7}{\left(x - 3\right)^{2}}$
The result of the chain rule is:

$- \frac{7 \left(x - 3\right)}{\left(x - 3\right)^{2} \left(2 x + 1\right)}$
Now simplify:

$- \frac{7}{\left(x - 3\right) \left(2 x + 1\right)}$

The answer is:

$- \frac{7}{\left(x - 3\right) \left(2 x + 1\right)}$

The graph

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The first derivative [src]

        /  2     2*x + 1 \
(x - 3)*|----- - --------|
        |x - 3          2|
        \        (x - 3) /
--------------------------
         2*x + 1

$$\frac{\left(x - 3\right) \left(\frac{2}{x - 3} - \frac{2 x + 1}{\left(x - 3\right)^{2}}\right)}{2 x + 1}$$

Simplify

The second derivative [src]

/    1 + 2*x\ /    1         2   \
|2 - -------|*|- ------ - -------|
\     -3 + x/ \  -3 + x   1 + 2*x/
----------------------------------
             1 + 2*x

$$\frac{\left(2 - \frac{2 x + 1}{x - 3}\right) \left(- \frac{2}{2 x + 1} - \frac{1}{x - 3}\right)}{2 x + 1}$$

Simplify

4-я производная [src]

  /    1 + 2*x\ /      1           8                 4                     2         \
6*|2 - -------|*|- --------- - ---------- - ------------------- - -------------------|
  \     -3 + x/ |          3            3            2                              2|
                \  (-3 + x)    (1 + 2*x)    (1 + 2*x) *(-3 + x)   (1 + 2*x)*(-3 + x) /
--------------------------------------------------------------------------------------
                                       1 + 2*x

$$\frac{6 \left(2 - \frac{2 x + 1}{x - 3}\right) \left(- \frac{8}{\left(2 x + 1\right)^{3}} - \frac{4}{\left(x - 3\right) \left(2 x + 1\right)^{2}} - \frac{2}{\left(x - 3\right)^{2} \left(2 x + 1\right)} - \frac{1}{\left(x - 3\right)^{3}}\right)}{2 x + 1}$$

Simplify

The third derivative [src]

  /    1 + 2*x\ /    1           4                2         \
2*|2 - -------|*|--------- + ---------- + ------------------|
  \     -3 + x/ |        2            2   (1 + 2*x)*(-3 + x)|
                \(-3 + x)    (1 + 2*x)                      /
-------------------------------------------------------------
                           1 + 2*x

$$\frac{2 \left(2 - \frac{2 x + 1}{x - 3}\right) \left(\frac{4}{\left(2 x + 1\right)^{2}} + \frac{2}{\left(x - 3\right) \left(2 x + 1\right)} + \frac{1}{\left(x - 3\right)^{2}}\right)}{2 x + 1}$$

Simplify

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Derivative of ln((2x+1)/(x-3))

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